Electric dipole moments of neutron and heavy quarks in the B-L symmetric MSSM

The searching for the electric dipole moments (EDMs) of neutron ($d_n$), $b$ quark ($d_b$) and $c$ quark ($d_c$) gives strict upper bounds on these quantities. And recently, new upper bounds on $d_b$, $d_c$ are obtained by the strict limit on $d_n$. The models of new physics (NP) with additional CP-violating (CPV) sources are constrained strictly by these EDMs. In this work, we focus on the CPV effects on these EDMs in the minimal supersymmetric extension (MSSM) of the SM with local $B-L$ gauge symmetry (B-LSSM). The contributions from one-loop and some two-loop diagrams to the quark EDM are given in general form, which can also be used in the calculation of quark EDM in other models of NP. Considering the constrains from updated experimental data, the numerical results show that the two-loop corrections can make important contributions to these EDMs. Compared with the MSSM, the effects of new CPV phases and new parameters in the B-LSSM on these EDMs are also explored.


I. INTRODUCTION
In the standard model (SM), the only sources of CP-violating (CPV) are the Cabbibo-Kobayashi-Maskawa (CKM) phases, which appears to be the origin of the CPV phenomena observed in nondiagonal processes involving the K and B mesons [1][2][3]. However, the observed baryon asymmetry of the universe indicates that the CPV sources in the SM are not sufficient, and new CPV sources are needed to generate the observed baryon asymmetry.
In addition, it is well known that the theoretical predictions of electric dipole moments (EDMs) in the SM are too tiny to be detected in the near future. When new CPV phases are introduced, the theoretical predictions of EDMs can be enhanced vastly, hence the EDM of elementary particle is a clear signal of CPV [4][5][6][7][8]. As a result, studying the EDMs of elementary particles is of prime importance, and measurements on the EDMs of neutron or fundamental particles provide sensitive approaches to investigate potential new sources of CPV. So far no EDM for the neutron, b quark or c quark has been detected, but strong bounds on these quantities have been obtained [9][10][11][12][13] |d n | < 3.0 × 10 −26 e · cm, |d b | < 2.0 × 10 −17 e · cm, |d c | < 4.4 × 10 −17 e · cm. (1) In addition, the chromo-EDM (CEDM) of heavy quark is constrained by the strict limit on the neutron EDM. The EDM of neutron can be expressed in terms of fundamental dipoles [14] d n = (1 ± 0.5)[1.4(d γ d − 0.25d γ u ) + 1.1e(d g d + 0.5d g u )] ± (22 ± 10) MeVC 5 , where d γ q , d g q , C 5 denote the quark EDM of q from the electroweak interaction, the CEDM of q and the coefficient of Weinberg operator at the chirality scale respectively. Using the running from Refs. [15,16] at one-loop, d γ,g d,u and C 5 can be expressed in terms of d g c at the scale m c [13]. Combining with the limits on d n , new upper limit on d g c is obtained [13], |d g c | < 1.0×10 −22 cm. Then assuming constructive interference between the EDM and CEDM contributions at the NP scale [17], new bounds on the EDM of b and c quark are derived by using the stringent limits on d g b in Ref. [18] and d g c in Ref. [13] |d b | < 1.2 × 10 −20 e · cm, which improves the previous ones in Eq. (1) by about three orders of magnitude. Since the experimental upper bounds on these quantities are very small, the contributions from new CPV phases are limited strictly by the present experimental data, hence researching NP effects on these EDMs may shed light on the mechanism of CPV.
In extensions of the SM, the supersymmetry is considered as one of the most plausible candidates. For the explanation of baryon asymmetry, electroweak baryogenesis (EWB) is one of the most well-known mechanisms, and new CPV phases are needed to enhance the asymmetry in this case. In Refs. [19][20][21][22][23], EWB is discussed in the MSSM in great detail, and the results show that the µ term (the bilinear Higgs mass term in the superpotential) is the dominant source of baryon asymmetry. However, when the phase of µ is taken to be large, the theoretical predictions of EDMs are several magnitudes larger than the corresponding upper bounds. The effects have been explored in Refs. [24][25][26][27][28][29][30][31][32]. The results show that the most interesting possibility to suppress these EDMs to below the corresponding experimental upper bounds is the contributions from different phases cancel each other. The CPV characters in supersymmetry are very interesting and studies on them may shed some light on the general characteristics of the supersymmetric model.
In this work, we explore the CPV effects on the EDM of neutron d n , b quark d b and c quark d c in the MSSM with local B − L gauge symmetry (B-LSSM) [33][34][35][36]. The model is based on the gauge symmetry group for the baryon number and L stand for the lepton number respectively. Compared with the MSSM, there are much more candidates for the dark matter [37][38][39][40] in the B-LSSM, which also accounts elegantly for the existence and smallness of the left-handed neutrino masses.
Since the exotic singlet Higgs and right-handed (s) neutrinos [41][42][43][44][45][46] releases additional parameter space from the LHC constraints, the model alleviates the little hierarchy problem of the MSSM [47]. In addition, the invariance under U(1) B−L gauge group imposes the R-parity conservation, which is assumed in the MSSM to avoid proton decay. And R-parity conservation can be maintained if U(1) B−L symmetry is broken spontaneously [48].
The paper is organized as follows. In Sec. II, the main ingredients of the B-LSSM are summarized briefly by introducing the superpotential and the general soft breaking terms.
Then the analysis on the EDM of neutron d n , b quark d b and c quark d c are presented in Sec. III. In order to see the corrections to these EDMs clearly, the numerical results of d n , Then in analogy to the ratio of the MSSM VEVs (tan β = v 2 v 1 ), we can define tan β ′ = u 2 u 1 . In addition, the superpotential of the B-LSSM can be written as where i, j are generation indices. Correspondingly, the soft breaking terms of the B-LSSM are generally given as

III. THE EDMS OF NEUTRON AND HEAVY QUARKS
For the neutron EMD d n , we adopt the values 0.5 and 12 MeV for the coefficients 1 ± 0.5 and 22 ± 10 MeV in Eq. (2) respectively, in order to coincide with the discussion in Ref. [13].
In addition, the quark EDM at the low scale can be written as where Λ χ = m q denotes the chirality breaking scale, m q denotes the corresponding quark mass. The Wilson coefficient of the purely gluonic Weinberg operator originates from the two-loop "gluino-squark" diagrams, and the concrete expression of C 5 can be written as [59][60][61]] where Λ denotes the matching scale, Zt(Zb) is the diagonalizing matrix for the squared mass matrix of stop (sbottom), and the function H can be found in Refs. [59][60][61].
Meanwhile, d γ q , d g q and C 5 are evolved with the renormalization group equations from the matching scale Λ down to the chirality breaking scale Λ χ [62][63][64][65] according to The effective Lagrangian for the quark EDMs can be written as where σ µν = i[γ µ , γ ν ]/2, q is the wave function for quark, and F µν is the electromagnetic field strength. Adopting the effective Lagrangian approach, the quark EDMs can be written where C L,R 2,6 represent the Wilson coefficients of the corresponding operators O L,R Similarly, the effective Lagrangian for the quark CEDMs can be written as where G µν is the SU(3) gauge field strength, T a is the SU(3) generators. Then the quark CEDMs can be written as where C L,R 7,8 represent the Wilson coefficients of the corresponding operators O L,R The one-loop Feynman diagrams contributing to the above amplitudes are depicted in Fig. 1. Calculating the Feynman diagrams, d γ q and d g q at the one-loop level can be written as The two-loop diagrams which contributes to d γ q and d g q are obtained by attaching a photon and a gluon respectively to the internal particles in all possible ways.
where x i denotes m 2 i /m 2 W , g 3 is the strong coupling constant, C L,R abc denotes the constant parts of the interaction vertex about abc which can be obtained through SARAH, and a, b, c denote the interacting particles. In order to see the contributions to EDMs introduced by the B-LSSM in addition to those already present in the MSSM, all constants C L,R abc appeared in our calculation are collected in appendix B. The functions I 1,2,3 can be written as The two-loop gluino corrections to the Wilson coefficients from the self-energy diagrams for quarks are considered, the corresponding Feynman diagrams are depicted in Fig. 2. The corresponding diploe moment diagrams are obtained by attaching a photon or gluon to the internal particles in all possible ways. Then the contributions from these two-loop diagrams to d γ q and d g q can be written as where the concrete expressions for the functions F 3,4,5 can be found in Ref. [66]. The similar expressions of two-loop gluino contributions can be found in Ref. [66]. We translate them into our notations which is written in general form and can be used in the calculation of quark EDM in other models of NP.
We should note that there are infrared divergencies in Fig. 2 when the SM quarks appear as internal particles, because we calculate these diagrams by expanding the external momentum. In this case, matching full theory diagrams to the corresponding two-loop diagrams in Fig. 2 is needed to cancel the infrared divergency. Taking Fig. 2(a) as example to illustrate how to cancel the infrared divergency, the corresponding diagrams are shown in Fig. 3. When the external gluon is attached to an internal particle in Fig. 2(a), and the external gluon can be attached to the same internal particle in Fig. 3(a) or (b). Then infrared divergency in the diagram by attaching a gluon in Fig. 2(a) can be cancelled by subtracting the corresponding diagram by attaching a gluon in the same way in Fig. 3.
In addition, the two-loop Barr-Zee type diagrams can also make contributions to the  Fig. 4. Then, the contributions from these two-loop Barr-Zee type diagrams to d γ q are given by [67] where s w ≡ sin θ W , c w ≡ cos θ W , and θ W is the Weinberg angle, T 3q denotes the isospin of the corresponding quark, the functions f γH , f ZH , f W W can be found in Ref. [67].

IV. NUMERICAL ANALYSIS
In this section, we present the numerical results of the EDMs gives us an upper bound on the ratio between the Z ′ mass and its gauge coupling at 99% CL as M Z ′ /g B > 6 TeV [70,71]. Experimental data collected at LEP also have been used to obtain upper bound on the Z −Z ′ mixing angle θ W ′ [72,73]. The bound is model-dependent, but current data from collider yields sin θ W ′ < ∼ 10 −3 for standard GUT models. The upper bound on Z − Z ′ mixing angle is considered in the following analysis. In addition, the LHC experimental data also constrain tan β ′ < 1.5. Since the contributions from heavy  which is needed to generate the baryon asymmetry can be cancelled by appropriate θ 3 . The contributions to d n from θ µ can be cancelled by the one of θ 3 results from the fact that θ µ affects the numerical results mainly through the squark mass matrixes, but the effects of it are suppressed by the corresponding Yukawa coupling constants which can be seen in Eq. (A1, A3). On the other hand, θ 3 appears in the analysis result directly and there is no suppressive factor. It can also explain that the contributions to d b from θ µ are comparable with the ones from θ 3 while the contributions to d n,c from θ µ are smaller than the ones from θ 3 , which can be seen directly by comparing Fig. 5(a), (c), (e) with Fig. 5(b), (d), (f).
We take M 3 , µ > ∼ 0.3 TeV in order to avoid the ranges excluded by the experiments [10]. Since both the two-loop gluino and Barr-Zee type diagrams can make important contributions to the numerical results, we take the summing of all contributions from these two-loop and one-loop corrections in the following analysis. Then d n versus M 3 , µ are plotted in Fig. 6(a), (b) Fig. 6(c, d), (e, f). Fig. 6(a, c, e) show that µ, θ 3 , θ µ affect the numerical results more obviously with large M 3 . Especially when TeV, the theoretical predictions of these EDMs can be larger than the corresponding results shown in Fig. 5 by about two orders of magnitude. Comparing with the results shown in Fig. 6(b, d, f), these parameters affect the numerical results more obviously when µ is large, but the trend is less obvious than the results shown in Fig. 6(a, c, e). It indicates that M 3 affect the numerical results more obvious than µ, and the contributions to d n from θ µ can be cancelled by appropriate θ 3 which is shown in Fig. 6(a, b). In addition, the experimental upper bounds on d b , d c also limit the parameter space when M 3 , µ are large as we can note in Fig. 6(c, d, We take |A Q | < 2 TeV approximately in order to satisfy the Color or Charge Breaking bounds [76]. Then d n versus A Q are plotted in Fig. 7  From Fig. 7(b), (d), (f) we can see that θ A can also make important contributions to these EDMs, and the signs of d n,b,c can be changed when we take the sign of A Q be opposite.
In addition, Fig. 7 Fig. 8(c, d), (e, f) respectively. There are no blue dashed lines in Fig. 8(c, d), (e, f) because the upper bounds on d b , d c are much larger than the results shown in the picture. On the basis (d L ,d R ), new definition of the mass matrix for down type squarks is given by which can be diagonalized by Zd.
On the basis (ũ L ,ũ R ), new definition of the mass matrix for up type squarks is given by where, which can be diagonalized by Zũ.